First of all, I am beginner for Fourier series I am trying to learn it.
I have a function $f(x) \in L^2 \left( \left[\pi,\pi \right] \right) $ and its Fourier Series is $$\sum ^{\infty }_{n=0}\dfrac{1}{n^{3}}\sin nx$$
My question is how can I say $f$ is periodic and continuous.
$\bullet$ for periodic part, I guess it is from definition. A Fourier series is an expansion of a periodic function $f$ in terms of an infinite sum of sines and cosines.
Any help would be appreciated.
The terms are dominated by those of $\sum \frac 1 {n^{3}}$ so the Fourier series converges uniformly to some function $g$ which is continuous since uniform limits if continuous functions are continuous. The partial sums converge to $f$ in $L^{2}$ and to $g$ pointwise which implies $f=g$ a.e. Hence, $f$ is almost everywhere equal to a continuous function.